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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

Characteristics of f, f', f''

         

Let \(f\) be a function such that \(f(0)=50.\) If the above graph represents the graph of the first derivative of \(f\) on the interval \(0<x<3,\) which of the following is the correct description of \(f\) on this interval?

For the function \(y=x^3+x^2-x-1,\) determine the interval where \(y\) is decreasing.

Suppose \(f\) is a function such that \(f(0)=33\) and the first derivative of \(f\) on the interval \(-2 < x < 2\) is as shown in the above diagram. Which of the following is the correct description of the function \(f\) on this interval?

Suppose \(f\) is a function defined on the closed interval \(-3 \le x \le 4\) with \(f(0)=42,\) such that the graph of \(f',\) the derivative of \(f,\) on the interval is as shown in the above diagram. On what intervals is \(f\) increasing?

Suppose \(f\) is a function defined on the closed interval \(-3 \le x \le 4\) with \(f(0)=3\) such that the graph of \(f',\) the derivative of \(f,\) on the interval is as shown in the above diagram. Find the \(x\)-coordinates of the points of inflection of \(f.\)

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